Trying to understand VGG convolution neural networks architecture
Good questions. Let me reply one by one.

1- Filter size can be increased. There is no limit for it. However, think two cases:


DNN part. Shape will be 1024 x 7 x 7 and it will map to 4096 features, it will cause 204M parameters at dense_1 layer. This change will cause two possible problem. Overfitting and inference/training speed.
Sparsity. You can implement 1024 features at 5th conv block and train the network and check the accuracy to see if increased feature count will make any impact or not.


So, to decide the number of con layers, it is a good practice to check sparsity of feature layers. 

2- If I did not understand your question wrong, those layers are DNN classifier as a fully connected layer.

3- Maxpooling is not causing us to see a subset of the image. It is just an algorithm to resizing. What you see in that video is the filter response to some objects exist in the image. In deeper level, do not expect to have visual shape like in shallow layers. You may even get single white pixel at deepest layer and it may be a response to an object like human face.